I want to show that if $\alpha$ and $\beta$ are rationally independent irrational numbers i.e. $\forall m,n \in\mathbb{Z}$ , $m\alpha + n\beta \not\in\mathbb{Z}$ , then the sequence
$\{ (n\alpha$ (mod 1) , $n\beta$ (mod 1) $\}_{n\in\mathbb{Z}}$ is dense in 2-Torus.
I managed to argue that $\{n\alpha$ (mod 1) $\}_{n\in\mathbb{Z}}$ and $\{n\beta$ (mod 1) $\}_{n\in\mathbb{Z}}$ are dense in 1-Torus. However, I do not see how to use rational independence to arrive at denseness of the above sequence in two torus.
The set $\{(k\alpha,k\beta) \mod 1: k \in \mathbb Z\}$ is a subgroup of $\mathbb T^2$, and its closure is a closed subgroup. If that is not all of $\mathbb T^2$, there must be a nontrivial member of the dual group $\mathbb Z^2$ that annihilates it, i.e. a pair of integers $(m,n)$, not both $0$, such that $m\alpha + n\beta \in \mathbb Z$, which would say $\alpha$ and $\beta$ are rationally dependent.